Discrepancy between Constant Properties Model and Temperature-Dependent Material Properties for Performance Estimation of Thermoelectric Generators

Abstract

The efficiency of a thermoelectric (TE) generator for the conversion of thermal energy into electrical energy can be easily but roughly estimated using a constant properties model (CPM) developed by Ioffe. However, material properties are, in general, temperature (T)-dependent and the CPM yields meaningful estimates only if physically appropriate averages, i.e., spatial averages for thermal and electrical resistivities and the temperature average (TAv) for the Seebeck coefficient (α), are used. Even though the use of αTAv compensates for the absence of Thomson heat in the CPM in the overall heat balance, we find that the CPM still overestimates performance (e.g., by up to 6% for PbTe) for many materials. The deviation originates from an asymmetric distribution of internally released Joule heat to either side of the TE leg and the distribution of internally released Thomson heat between the hot and cold side. The Thomson heat distribution differs from a complete compensation of the corresponding Peltier heat balance in the CPM. Both effects are estimated quantitatively here, showing that both may reach the same order of magnitude, but which one dominates varies from case to case, depending on the specific temperature characteristics of the thermoelectric properties. The role of the Thomson heat distribution is illustrated by a discussion of the transport entropy flow based on the α(T) plot. The changes in the lateral distribution of the internal heat lead to a difference in the heat input, the optimum current and thus of the efficiency of the CPM compared to the real case, while the estimate of generated power at maximum efficiency remains less affected as it is bound to the deviation of the optimum current, which is mostly <1%. This deviation can be corrected to a large extent by estimating the lateral Thomson heat distribution and the asymmetry of the Joule heat distribution. A simple guiding rule for the former is found.

Thermoelectric degrees of freedom determining thermoelectric efficiency

“…Even using the ignoring τ and β can be a good estimation, showing why the CPM can work well for some cases while failing for other cases, as previously reported ( Sherman et al. (1960) ; Ponnusamy et al. (2020a , 2020b) ).…”

“…Its root mean squared relative error (RMSRE) is only 1.9% for 277 published materials; see STAR Methods (maximum efficiency prediction using h gen max ), Figure S3; Tables S1-S6 and S9). Even using the h gen max ðZ ð0Þ gen ; 0; 0Þ ignoring t and b can be a good estimation, showing why the CPM can work well for some cases while failing for other cases, as previously reported (Sherman et al (1960); Ponnusamy et al (2020aPonnusamy et al ( , 2020b). In the same figure, our approximation method is compared to other figure of merit models: the peak zT, the ½ZT mod of Min et al (Min et al (2004)) and the engineering ½ZT eng generic formula of Kim et al (Kim et al (2015a)).…”

“…(1960) ; Sunderland and Burak (1964) ; Kim et al. (2015a , 2015b) ; Wee (2011) ; Ponnusamy et al. (2020a) ).…”

“…But they are still inaccurate owing to a poor approximation of temperature gradient and/or an overestimation of Thomson heat ( Ponnusamy et al. (2020b , 2020a) ). While integrating or z over T is one of the most widely used averaging schemes, it severely overestimates the ( Kim et al.…”

“…However, the zT inaccurately predicts the efficiency because the thermoelectric properties greatly vary with T (Goupil (2015); Borrego Larralde (1961); Borrego (1964); Sherman et al (1960); Sunderland and Burak (1964); Kim et al (2015aKim et al ( , 2015b; Wee (2011); Ponnusamy et al (2020a)). To improve the accuracy, various single parameters averaging the zT have been developed: the effective figure of merit z eff : = CaD 2 CrkD by Ioffe (Ioffe (1957)) and Borrego (Borrego Larralde (1961)), the engineering ZT (½ZT eng : = CaD 2 CrDCkD DT) by Kim et al (Kim et al (2015a)), and the modified figure of merit (½ZT mod ) by Min et al (Min et al (2004)), where the bracket C ,D indicates averaging over T. But they are still inaccurate owing to a poor approximation of temperature gradient and/or an overestimation of Thomson heat (Ponnusamy et al (2020b(Ponnusamy et al ( , 2020a). While integrating zT or z over T is one of the most widely used averaging schemes, it severely overestimates the h max…”